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<title>Figure 6.24: Fitting a convex function to given data</title>
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<h1>Figure 6.24: Fitting a convex function to given data</h1>
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<span class="comment">% Section 6.5.5</span>
<span class="comment">% Boyd &amp; Vandenberghe "Convex Optimization"</span>
<span class="comment">% Original by Lieven Vandenberghe</span>
<span class="comment">% Adapted for CVX by Argyris Zymnis - 11/27/2005</span>
<span class="comment">%</span>
<span class="comment">% Here we find the convex function f that best fits</span>
<span class="comment">% some given data in the least squares sense.</span>
<span class="comment">% To do this we solve</span>
<span class="comment">%     minimize    ||yns - yhat||_2</span>
<span class="comment">%     subject to  yhat(j) &gt;= yhat(i) + g(i)*(u(j) - u(i)), for all i,j</span>

clear

<span class="comment">% Noise level in percent and random seed.</span>
rand(<span class="string">'state'</span>,29);
noiseint=.05;

<span class="comment">% Generate the data set</span>
u = [0:0.04:2]';
m=length(u);
y = 5*(u-1).^4 + .6*(u-1).^2 + 0.5*u;
v1=u&gt;=.2;
v2=u&lt;=.6;
v3=v1.*v2;
dipvec=((v3.*u-.4*ones(1,size(v3,2))).^(2)).*v3;
y=y+40*(dipvec-((.2))^2*v3);

<span class="comment">% add perturbation and plots the input data</span>
randf=noiseint*(rand(m,1)-.5);
yns=y+norm(y)*(randf);
figure
plot(u,yns,<span class="string">'o'</span>);

<span class="comment">% min. ||yns-yhat||_2</span>
<span class="comment">% s.t. yhat(j) &gt;= yhat(i) + g(i)*(u(j) - u(i)), for all i,j</span>
cvx_begin
    variables <span class="string">yhat(m)</span> <span class="string">g(m)</span>
    minimize(norm(yns-yhat))
    subject <span class="string">to</span>
        yhat*ones(1,m) &gt;= ones(m,1)*yhat' + (ones(m,1)*g').*(u*ones(1,m)-ones(m,1)*u');
cvx_end

nopts =1000;
t = linspace(0,2,nopts);
f = max(yhat(:,ones(1,nopts)) + <span class="keyword">...</span>
      g(:,ones(1,nopts)).*(t(ones(m,1),:)-u(:,ones(1,nopts))));
plot(u,yns,<span class="string">'o'</span>,t,f,<span class="string">'-'</span>);
axis <span class="string">off</span>
<span class="comment">%print -deps interpol_convex_function2.eps</span>
</pre>
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<pre class="codeoutput">
 
Calling Mosek 9.1.9: 2602 variables, 103 equality constraints
   For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 103             
  Cones                  : 1               
  Scalar variables       : 2602            
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 103             
  Cones                  : 1               
  Scalar variables       : 2602            
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 100
Optimizer  - Cones                  : 1
Optimizer  - Scalar variables       : 2502              conic                  : 52              
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00              dense det. time        : 0.00            
Factor     - ML order time          : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 3874              after factor           : 5050            
Factor     - dense dim.             : 0                 flops                  : 3.71e+05        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   1.0e+00  7.9e+00  2.0e+00  0.00e+00   0.000000000e+00   -1.000000000e+00  1.0e+00  0.00  
1   4.0e-01  3.2e+00  1.0e+00  -8.04e-01  -2.411440521e+00  -2.370632537e+00  4.0e-01  0.01  
2   9.9e-02  7.8e-01  2.2e-01  -2.37e-01  -4.901628796e+00  -4.291091481e+00  9.9e-02  0.01  
3   2.1e-02  1.6e-01  2.2e-02  6.31e-01   -5.225708045e+00  -5.095297522e+00  2.1e-02  0.01  
4   3.6e-03  2.9e-02  1.5e-03  1.01e+00   -5.317798571e+00  -5.301114814e+00  3.6e-03  0.02  
5   5.2e-04  4.1e-03  8.0e-05  1.04e+00   -5.192975323e+00  -5.190743539e+00  5.2e-04  0.02  
6   3.9e-04  3.1e-03  5.4e-05  9.00e-01   -4.828858712e+00  -4.826977695e+00  3.9e-04  0.02  
7   3.5e-04  2.7e-03  4.6e-05  7.61e-01   -4.632061026e+00  -4.630190100e+00  3.5e-04  0.02  
8   3.0e-04  2.4e-03  3.8e-05  8.93e-01   -4.463375636e+00  -4.461799341e+00  3.0e-04  0.02  
9   2.4e-04  1.9e-03  2.8e-05  7.14e-01   -4.075779742e+00  -4.074227721e+00  2.4e-04  0.02  
10  1.7e-04  1.4e-03  1.7e-05  9.03e-01   -3.762806997e+00  -3.761754840e+00  1.7e-04  0.02  
11  1.1e-04  8.9e-04  9.6e-06  7.52e-01   -3.228831403e+00  -3.227990075e+00  1.1e-04  0.02  
12  6.5e-05  5.2e-04  4.2e-06  9.55e-01   -2.974940962e+00  -2.974487008e+00  6.5e-05  0.02  
13  3.2e-05  2.5e-04  1.5e-06  8.68e-01   -2.579655374e+00  -2.579395159e+00  3.2e-05  0.02  
14  1.4e-05  1.1e-04  4.6e-07  9.99e-01   -2.471441621e+00  -2.471322262e+00  1.4e-05  0.02  
15  3.8e-06  3.0e-05  6.3e-08  9.75e-01   -2.322286314e+00  -2.322251513e+00  3.8e-06  0.03  
16  1.2e-06  9.2e-06  1.1e-08  1.00e+00   -2.291714195e+00  -2.291703000e+00  1.2e-06  0.03  
17  1.3e-07  1.0e-06  3.9e-10  9.99e-01   -2.276071639e+00  -2.276070399e+00  1.3e-07  0.03  
18  1.3e-08  1.0e-07  1.2e-11  1.00e+00   -2.274446781e+00  -2.274446656e+00  1.3e-08  0.03  
19  3.8e-10  3.0e-09  6.4e-14  1.00e+00   -2.274271282e+00  -2.274271278e+00  3.8e-10  0.03  
Optimizer terminated. Time: 0.03    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -2.2742712819e+00   nrm: 1e+00    Viol.  con: 3e-13    var: 1e-09    cones: 0e+00  
  Dual.    obj: -2.2742712781e+00   nrm: 6e+01    Viol.  con: 0e+00    var: 1e-08    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.03    
    Interior-point          - iterations : 19        time: 0.03    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 0         time: 0.00    
        Dual                - iterations : 0         time: 0.00    
        Clean primal        - iterations : 0         time: 0.00    
        Clean dual          - iterations : 0         time: 0.00    
    Simplex                 -                        time: 0.00    
      Primal simplex        - iterations : 0         time: 0.00    
      Dual simplex          - iterations : 0         time: 0.00    
    Mixed integer           - relaxations: 0         time: 0.00    

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +2.27427
 
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